Attractive Quantum Subsystems and Feedback-Stabilization Problems
Lorenza Viola, Francesco Ticozzi · 2007
We propose a general theoretical framework that is suitable to study a wide class of stabilization problems for quantum Markovian dynamical systems. Building on systemtheoretic ideas, we propose definitions of invariant and attractive quantum subsystem, characterize Markovian invariance properties, and provide sufficient conditions for attraction. The general framework and results are illustrated by addressing the potential of output-feedback Markovian control strategies for quantum pure state-stabilization. In particular, constructive results for the synthesis of stabilizing semigroups in arbitrary finite-dimensional Markovian systems are established. I. BACKGROUND AND MOTIVATIONS Stabilization problems are of central relevance for many quantum control applications, ranging from state preparation of quantum-optical and nano-mechanical systems to generation of noise-protected realizations of quantum information in realistic devices [1]. Dynamical systems undergoing Markovian evolution [2], [3] are both widely relevant from a physical standpoint and present distinctive control challenges – preventing, in particular, open-loop quantum-engineering and stabilization methods based on dynamical decoupling to be viable [4], [5]. However, we show here how a wide class of stabilization problems can be effectively treated in a general framework, provided by attractive quantum subsystems. After introducing the main ideas and definitions along with some general results, we shall explore their application to pure-state preparation problems for Markovian output-feedback control. We refer to the forthcoming journal version of the present paper [6] for detailed proofs we shall omit or merely sketch in the following sections. Consider a separable Hilbert space H over the complex field C. Let B(H) represent the set of linear bounded operators on H, H(H) denoting the real subspace of Hermitian operators, with I, O being the identity and the zero operator, respectively. In the standard statistical formulation of quantum mechanics [7], [8], the dimension of the Hilbert space H associated with the quantum system of interest, Q, is determined by the physics of the problem. In what follows, we consider finite-dimensional systems, i.e. dim(H) 0 being described by a TracePreserving, Completely-Positive (TPCP) map Tt(·) [12], [1]. A differential equation for the density operator of I may be derived provided that a forward composition law holds: Definition 1 (QDS): A quantum dynamical semigroup is a one-parameter family of TPCP maps {Tt(·), t ≥ 0} that satisfies: (i) T0 = I, (ii) Tt ◦ Ts = Tt+s, ∀t, s > 0, (iii) trace(Tt(ρ)X) is a continuous function of t, ∀ρ ∈ D(HI), ∀X ∈ B(HI). Due to the trace and positivity preserving assumptions, a QDS is a semigroup of contractions. It has been proved [10], [13] that the Hille-Yoshida generator for a QDS exists and can be cast in the following canonical form: